Abstract: Potential functions are a standard tool in the design and analysis of online learning algorithms. However, the best choice of potential function remained unclear. This talk starts with early work [Cesa-Bianchi et al 1996] that identified a min-max optimal potential function for the problems of online {0,1} predictions where the best expert is known to make at most $k$ mistakes. We extend the analysis to DTOL and derive a backwards recursion for time-dependent potentials. We extend the game to continuous time and show that for any potential function with strictly positive four derivatives the min/max optimal adversarial strategy is Brownian motion. The min/max optimal learner strategy is the best response to Brownian motion. Using this analysis the Normal-Hedge potential [Chaudhuri et al 2009, Luo, Schapire 2015] provides an almost optimal bound for unknown horizon, and we derive second order bounds for easy sequences.

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